Allegri survived a turbulent season at the San Siro last season to help Milan secure Champions League qualification, overcoming a shaky start to finish third in Serie A.
Gattuso believes that Allegri should be high on the list of replacements when Prandelli's role with the national side comes to an end, while declaring an ambition to coach Italy himself in the future.
He told Sky Sport Italia: "Allegri has the capability to do well. Leading Italy is a great responsibility, but Max has shown that he is a very good tactician. He would do well if given the role.
"Am I aiming for the Italy job? For now I'm focused on taking Palermo into the top flight. Saying that, I, like Allegri, am targeting such a role but I have a long road to go down first. I could think about it one day though.
"I was in the Giro Azzurro for 11 years, I know what a happy Azzurro dressing room is and I'd like to breathe that air again."
The 35-year-old is currently in charge of Serie B team Palermo, a position he took up during the summer.
Allegri admitted earlier this week that he would love to take charge of Italy in the future, but for now remains focused on life at the San Siro.